Method for regulating a voltage or a current of an RLC filter, a recording medium and vehicles for this method

ABSTRACT

This deadbeat control method for regulating an output voltage U c  or an output current I l  of a low-pass RLC filter includes:
         calculation ( 92 ) of a current setting Ī uc  for the average intensity Ī u  of a DC current I u  flowing through a first output point of the filter between instants t i  and t i+1 , this setting Ī uc  being established from discretized state equations of the filter in such a way that the voltage U c  or the line current I l  is equal to a predetermined setting of voltage U cc  or of line current I lc  at the instant t i+1 ,   control ( 100 ) of an electric converter in order to produce a current I u  flowing through the filter, the average intensity Ī u  of which between the instants t i  and t i+1  is equal to the current setting Ī uc .

This application claims priority French application FR 07 03719, filed on May 25, 2007, the entire disclosure of which is incorporated by reference herein.

The present invention relates to a method for regulating a voltage or a current of an RLC filter, a recording medium and vehicles for this method.

BACKGROUND OF THE INVENTION

More specifically, the applicant is aware of methods for regulating a voltage U_(c) between a first and a second output point of a low-pass RLC filter of natural period T_(f), this RLC filter including two input points electrically connected, respectively, to the conductors of a DC bus of an electric vehicle powered via a catenary, the first and the second output points being electrically connected to a controllable electric converter for controlling the torque exerted by an electric traction motor of the electric vehicle, the stator time constant τ of this motor being strictly less than the natural period T_(f).

These regulation methods include the measurement or estimate of the intensity I_(li) of a line current I_(l) flowing through the inductance of the filter at an instant t_(i), of the voltage U_(ci) between the output points of the filter at the instant t_(i), and of a line voltage U_(l) between the input points of the filter.

The applicant is also aware of methods for regulating a line current I_(l) flowing through an inductance L of a low-pass RLC filter of natural period T_(f), this filter including:

-   -   two input points electrically connected, respectively, to the         conductors of a DC bus of an electric vehicle powered via a         catenary, and     -   first and second output points, the first and the second output         points being electrically connected to a controllable electric         converter in order to cause the torque of an electric traction         motor of the electric vehicle to vary, the stator time constant         τ of this electric motor being strictly less than the natural         period T_(f).

These methods include the measurement or estimate of the intensity I_(li) of the line current I_(l) at an instant t_(i), of a voltage U_(ci) between the output points of the filter at the instant t_(i) and of a line voltage U_(l) between the input points of the filter.

Here, the term “catenary” refers to both an overhead line against which a pantograph rubs in order to power the electric vehicle and a ground-based rail against which a contact shoe slides in order to power an electric vehicle. This ground-based rail is more often known by the term “third rail”.

The stator time constant τ of an electric motor is defined by the following relationship:

$\tau = \frac{L_{m\;}}{R_{m}}$ where:

L_(m) is the stator inductance of the electric motor, and

R_(m) is the stator resistance of the electric motor.

This time constant is typically between 4 ms and 200 ms for the electric traction motors of an electric vehicle.

The natural period T_(f) of the RLC filter is defined by the following formula: T _(f)=2π√{square root over (LC)}

This natural period T_(f) must be strictly greater than the time constant τ of the motor, otherwise the RLC filter cannot fulfil its function as a low-pass filter in relation to rapid variations in the current consumed or produced by the motor. Another purpose of the RLC filter is to reduce the source impedance, or the load impedance, as seen by the converter.

The time to speed up the electric motor is defined here as being the time required to cause its speed to vary by a significant fraction, for example 1/1000, of its maximum speed with its maximum torque.

In the known methods, the regulation process involves the use of a feedback loop to establish the difference between a voltage setting U_(cc) between the output points of the filter or a line current setting I_(lc) and a measured value. These methods operate correctly but do not provide for reacting quickly enough to sudden variations in the line voltage U_(l) or resistive torque of the motor. For example, these sudden variations in the line voltage U_(l) or resistive torque can arise:

-   -   if the pantograph becomes detached from the catenary, i.e. when         the pantograph loses mechanical and electrical contact with the         catenary,     -   if the pantograph becomes reattached to the catenary, i.e. when         the pantograph re-establishes mechanical and electrical contact         with the catenary, or     -   in the event of a loss of adhesion between the drive wheels of         the electric vehicle and the wheel bearing structures.

The invention aims to remedy these problems by proposing a quicker method for regulating the voltage U_(c) or the line current I_(l).

SUMMARY OF THE INVENTION

Therefore, a subject of a invention is a deadbeat control method for regulating the voltage U_(c), in which the method includes:

-   -   calculation of a current setting Ī_(uc) for the average         intensity Ī_(u) of a DC current I_(u) flowing through the first         output point of the filter between the instant t_(i) and an         instant t_(i+1), this setting Ī_(uc) being established from         discretized state equations of the filter in such a way that the         voltage U_(c) is equal to a predetermined voltage setting U_(cc)         at the instant t_(i+1), these discretized state equations         between them establishing relationships between the intensities         I_(li) and I_(l,i+1) of the line current I_(l) at the instants         t_(i) and t_(i+1) respectively, the voltages U_(ci) and         U_(c,i+1) between the output points of the filter at the         instants t_(i) and t_(i+1) respectively, the average line         voltage Ū_(l) between the instants t_(i) and t_(i+1) and the         average intensity Ī_(u),     -   control of the electric converter in order to produce a current         I_(u) flowing through the output point of the filter, the         average intensity Ī_(u) of which between the instants t_(i) and         t_(i+1) is equal to the current setting Ī_(uc), the time         interval T between the instants t_(i) and t_(i+1) being strictly         less than 5τ.

Another subject of the invention is a deadbeat control method for regulating the intensity of the line current I_(l), in which the method includes:

-   -   calculation of a current setting Ī_(uc) for the average         intensity Ī_(u) of a DC current I_(u) flowing through the first         output point of the filter between the instant t_(i) and an         instant t_(i+1), this setting Ī_(uc) being established from         discretized state equations of the filter in such a way that the         voltage U_(c) is equal to a predetermined voltage setting U_(cc)         at the instant t_(i+1), these discretized state equations         between them establishing relationships between the intensities         I_(li) and I_(l,i+1) of the line current I_(l) at the instants         t_(i) and t_(i+1) respectively, the voltages U_(ci) and         U_(c,i+1) between the output points of the filter at the         instants t_(i) and t_(i+1) respectively, the average line         voltage Ū_(l) between the instants t_(i) and t_(i+1) and the         average intensity Ī_(u),     -   control of the electric converter in order to produce a current         I_(u) flowing through the output point of the filter, the         average intensity Ī_(u) of which between the instants t_(i) and         t_(i+1) is equal to the setting Ī_(uc), the time interval T         between the instants t_(i) and t_(i+1) being strictly less than         5τ.

These “deadbeat control” methods provide for arriving at the setting from the next regulation instant t_(i+1). To this end, these methods do not implement a feedback loop.

More specifically, in the above deadbeat control methods, the average intensity setting Ī_(uc) is determined from discretized state equations of the RLC filter without using a feedback loop. These methods therefore provide for ensuring that the voltage U_(c) or the line current I_(l) has reached its setting exactly at the end of the interval T. Furthermore, since the interval T is in this case chosen to be small, i.e. less than five times the time constant τ of the motor, these methods react much more quickly than regulation methods that implement a feedback loop. Under these conditions, voltage surges of the line voltage U_(l) or current surges of the line current I_(l) are limited much more effectively in particular in the event of detachment or reattachment of the pantograph, or in the event of a loss of adhesion.

It will also be noted that adjustments to the average intensity Ī_(u) are made at intervals T that are less than 5τ. The intervals T are therefore much shorter than the time to speed up the electric motor due to the inertia of the electric vehicle brought to the motor shaft and due to the inertia of the rotor of the electric motor itself, such that they are not felt by the driver or passengers of this vehicle. They therefore also do not interfere with the method for controlling the torque of the electric motor.

The embodiments of the method for regulating the voltage U_(c) can include one or more of the following features:

-   -   the voltage setting U_(cc) is chosen always to be less than or         equal to a voltage limit U_(cmax), the voltage limit U_(cmax)         corresponding to the maximum voltage allowable at the input of         the electric converter or between the output points of the         filter;     -   construction of an estimate I_(lp) of the line current I_(l)         which will be reached if the voltage U_(c) is equal to the         voltage setting U_(cc) at the instant t_(i+1),     -   comparison of the estimate I_(lp) with at least one         predetermined line current limit I_(lm),     -   only if the predetermined line current limit I_(lm) is crossed,         modification of the voltage setting U_(cc) so as to obtain a         temporary voltage setting U_(ccm) which corresponds to a line         current estimate I_(lp) that does not cross the predetermined         line current limit I_(lm), and use of the temporary voltage         setting U_(ccm) in place of the voltage setting U_(cc) during         the calculation of the average current setting Ī_(uc) only for         the interval T in progress, and     -   if the predetermined limit I_(lm) is not crossed, use of the         voltage setting U_(cc) for calculating the average current         setting Ī_(uc) for the interval T in progress;     -   the setting Ī_(uc) is a solution of the following system of         equations:

$\begin{matrix} {{I_{1p} - {\mu_{2} \cdot C \cdot U_{cc}}} = {{{\mathbb{e}}^{\mu_{1} \cdot T} \cdot \left( {I_{li} - {\mu_{2} \cdot C \cdot U_{ci}}} \right)} + {a_{1} \cdot \left( {{\mu_{2} \cdot {\overset{\_}{I}}_{uc}} + {\frac{1}{L} \cdot {\overset{\_}{U}}_{1}}} \right)}}} \\ {{I_{1p} - {\mu_{1} \cdot C \cdot U_{cc}}} = {{{\mathbb{e}}^{\mu_{2} \cdot T} \cdot \left( {I_{li} - {\mu_{1} \cdot C \cdot U_{ci}}} \right)} + {a_{2} \cdot \left( {{\mu_{1} \cdot {\overset{\_}{I}}_{uc}} + {\frac{1}{L} \cdot {\overset{\_}{U}}_{1}}} \right)}}} \end{matrix}$ where:

R and L are the values of the resistance and the inductance, respectively, of the RLC filter, and which are connected in series between the first input and output points,

C is the capacitance of the capacitor connected between the first and second output points,

μ₁, and μ₂ are the eigenvalues of an evolution matrix and are defined by the following relationships:

$\begin{matrix} {\mu_{1} = \frac{{{- R} \cdot C} + \sqrt{{R^{2} \cdot C^{2}} - {4 \cdot L \cdot C}}}{2 \cdot L \cdot C}} \\ {\mu_{2} = \frac{{{- R} \cdot C} - \sqrt{{R^{2} \cdot C^{2}} - {4 \cdot L \cdot C}}}{2 \cdot L \cdot C}} \end{matrix}$

a₁, and a₂ are values defined by the following relationships:

$\begin{matrix} {a_{1} = \frac{{\mathbb{e}}^{\mu_{1} \cdot T} - 1}{\mu_{1}}} \\ {a_{2} = \frac{{\mathbb{e}}^{\mu_{2} \cdot T} - 1}{\mu_{2}}} \end{matrix}$ the voltage setting U_(cc) is constructed from the line voltage U_(l) in such a way that its power spectrum does not exhibit any harmonic beyond the frequency 0.9/T_(f).

The embodiments of the method for regulating the voltage U_(c) exhibit, furthermore, the following advantages:

-   -   by keeping the voltage U_(c) below the limit U_(cmax), untimely         trips of a safety device, such as a rheostatic chopper for         clipping any voltage surge on the DC bus, are avoided,     -   by using the temporary voltage setting U_(ccm), the intensity of         the line current I_(l) can be maintained within an acceptable         range while limiting variations in the voltage U_(c), and     -   by choosing a voltage setting U_(cc) such that its power         spectrum does not exhibit any harmonic beyond the frequency         0.9/T_(f), oscillations of the line current I_(l) and of the         capacitor voltage U_(c) close to the natural frequency of the         filter are avoided and also the control energy is minimized.

The embodiments of the method for regulating the intensity of the line current I_(l) can include one or more of the following features:

-   -   the line current setting I_(lc) is chosen always to be less than         or equal to a limit I_(lmax), the limit I_(lmax) corresponding         to the intensity of the line current I_(l) at which a         circuit-breaker of a power substation of the catenary or a         circuit-breaker of the vehicle is tripped;     -   the line current setting I_(lc) is chosen always to be greater         than or equal to a limit I_(lmin), the limit I_(lmin)         corresponding to the intensity of the line current below which         the inductance L of the filter is desaturated;     -   construction of an estimate U_(cp) of the voltage U_(c) which         will be reached between the output points at the instant t_(i+1)         if the intensity of the line current I_(l) is equal to the         setting I_(lc) at the instant t_(i+1),     -   comparison of the voltage estimate U_(cp) with at least one         predetermined voltage limit U_(cm),     -   only if the predetermined voltage limit U_(cm) is crossed,         modification of the line current setting I_(lc) so as to obtain         a temporary line current setting I_(lcm) which corresponds to an         estimate U_(cp) that does not cross the predetermined limit         U_(cm), and use of the temporary line current setting I_(lcm) in         place of the line current setting I_(lc) during the control of         the converter only over the interval T in progress, and     -   if the predetermined voltage limit U_(cm) is not crossed, use of         the line current setting I_(lc) during the calculation of the         average current setting Ī_(uc) for the interval T in progress;     -   the average current setting Ī_(uc) is a solution of the         following system of equations:

$\begin{matrix} {{I_{1c} - {\mu_{2} \cdot C \cdot U_{cp}}} = {{{\mathbb{e}}^{\mu_{1} \cdot T} \cdot \left( {I_{li} - {\mu_{2} \cdot C \cdot U_{ci}}} \right)} + {a_{1} \cdot \left( {{\mu_{2} \cdot {\overset{\_}{I}}_{uc}} + {\frac{1}{L} \cdot {\overset{\_}{U}}_{1}}} \right)}}} \\ {{I_{1c} - {\mu_{1} \cdot C \cdot U_{cp}}} = {{{\mathbb{e}}^{\mu_{2} \cdot T} \cdot \left( {I_{li} - {\mu_{1} \cdot C \cdot U_{ci}}} \right)} + {a_{2} \cdot \left( {{\mu_{1} \cdot {\overset{\_}{I}}_{uc}} + {\frac{1}{L} \cdot {\overset{\_}{U}}_{1}}} \right)}}} \end{matrix}$ where:

R and L are the values of the resistance and the inductance, respectively, of the RLC filter, and which are connected in series between the first input and output points,

C is the capacitance of the capacitor connected between the first and second output points,

μ₁ and μ₂ are the eigenvalues of an evolution matrix and are defined by the following relationships:

$\begin{matrix} {\mu_{1} = \frac{{{- R} \cdot C} + \sqrt{{R^{2} \cdot C^{2}} - {4 \cdot L \cdot C}}}{2 \cdot L \cdot C}} \\ {\mu_{2} = \frac{{{- R} \cdot C} - \sqrt{{R^{2} \cdot C^{2}} - {4 \cdot L \cdot C}}}{2 \cdot L \cdot C}} \end{matrix}$

a₁ and a₂ are values defined by the following relationships:

$\begin{matrix} {a_{1} = \frac{{\mathbb{e}}^{\mu_{1} \cdot T} - 1}{\mu_{1}}} \\ {a_{2} = \frac{{\mathbb{e}}^{\mu_{2} \cdot T} - 1}{\mu_{2}}} \end{matrix}$

-   -   a first phase for regulating only the voltage U_(c) between the         first and second output points of the filter,     -   a second phase for regulating only the line current I_(l),     -   switchover from the first phase to the second phase as soon as         the line current I_(l) crosses a limit I_(lm) and switchover         from the second phase to the first phase as soon as the line         current I_(l) crosses the same or another limit in the opposite         direction.

The embodiments of the method for regulating the line current I_(l) exhibit, furthermore, the following advantages:

-   -   by choosing the line current setting I_(lc) to be lower than the         limit I_(lmax), untimely trips of the circuit-breaker of a         substation, or of the circuit-breaker protecting equipment on         the vehicle itself, are always avoided,     -   by choosing the line current setting I_(lc) to be always greater         than the limit I_(lmin), the saturated inductance L is always         preserved and therefore this provides for remaining within a         linear operating zone, thereby avoiding a scenario in which the         inductance L suddenly releases a high amount of energy while         desaturating,     -   by using the temporary line current setting I_(lcm), the voltage         U_(c) is always maintained within an acceptable operating range         while limiting variations in the line current I_(l),     -   alternating between phases for regulating the voltage U_(c) only         and the line current I_(l) only provides for maintaining both         the voltage U_(c) and line current I_(l) within acceptable         operating ranges.

The methods for regulating the voltage U_(c) or the line current I_(l) can include one or more of the following features:

-   -   the method includes the control of a rheostat in order to         produce, in combination with the control of the converter, the         current I_(u) flowing through the first output terminal, the         average intensity Ī_(u) of which between the instants t_(i) and         t_(i+1) is equal to the current setting Ī_(uc);     -   the interval T is less than or equal to τ/5.

The above embodiments of methods for regulating the voltage U_(c) or the line current I_(l) exhibit, furthermore, the following advantages:

-   -   by using the rheostat to produce the current I_(u) in addition         to the converter, a variation in the voltage U_(c) or the line         current I_(l) can be compensated for more rapidly than if only         the converter were used,     -   by choosing the interval T to be less than or equal to τ/5, the         amplitude of variations in the current I_(u) can be limited,         thereby improving the characteristics of the regulation method.

Another subject of the invention is an information recording medium including instructions for executing any one of the above regulation methods, when these instructions are executed by an electronic computer.

Another subject of the invention is an electric vehicle including:

-   -   a DC bus formed by two conductors,     -   at least one traction motor of the electric vehicle having a         stator time constant τ,     -   a controllable electric converter intended to cause the torque         of the traction motor to vary,     -   a low-pass RLC filter including two input points electrically         connected, respectively, to the two conductors of the DC bus,         and first and second output points electrically connected to the         electric converter,     -   sensors or estimators intended to measure or estimate the         intensity I_(li) of a line current I_(l) flowing through the         inductance of the filter at the instant t_(i), a voltage U_(ci)         between the output points of the filter at the instant t_(i),         and a line voltage U_(l) between the input points of the filter;         -   a computer for calculating a current setting Ī_(uc) for the             average intensity Ī_(u) of a DC current I_(u) flowing             through the first output point of the filter between the             instant t_(i) and an instant t_(i+1), this current setting             Ī_(uc) being established from discretized state equations of             the filter in such a way that the voltage U_(c) is equal to             a predetermined voltage setting U_(cc) at the instant             t_(i+1), these discretized state equations between them             establishing relationships between the intensities I_(li)             and I_(l,i+1) of the line current I_(l) at the instants             t_(i) and t_(i+1) respectively, the voltages U_(ci) and             U_(c,i+1) between the output points of the filter at the             instants t_(i) and t_(i+1) respectively, the average line             voltage Ū_(l) between the instants t_(i) and t_(i+1) and the             average intensity Ī_(u),     -   a control unit for controlling the electric converter in order         to produce a current I_(u) flowing through the output point of         the filter, the average intensity Ī_(u) of which between the         instants t_(i) and t_(i+1) is equal to the current setting         Ī_(uc), the time interval T between the instants t_(i) and         t_(i+1) being strictly less than 5τ.

Another subject of the invention is another electric vehicle including:

-   -   a computer for calculating a current setting Ī_(uc) for the         average intensity Ī_(u) of a DC current I_(u) flowing through         the first output point of the filter between the instant t_(i)         and an instant t_(i+1), this current setting Ī_(uc) being         established from discretized state equations of the filter in         such a way that the intensity of the line current I_(l) is equal         to a predetermined line current setting I_(lc) at the instant         t_(i+1), these discretized state equations between them         establishing relationships between the intensities I_(li) and         I_(l,i+1) of the line current I_(l) at the instants t_(i) and         t_(i+1) respectively, the voltages U_(ci) and U_(c,i+1) between         the output points of the filter at the instants t_(i) and         t_(i+1) respectively, the average line voltage Ū_(l) between the         instants t_(i) and t_(i+1) and the average intensity Ī_(u), and     -   a control unit for controlling the electric converter in order         to produce a current I_(u) flowing through the output point of         the filter, the average intensity Ī_(u) of which between the         instants t_(i) and t_(i+1) is equal to the current setting         Ī_(uc), the time interval T between the instants t_(i) and         t_(i+1) being strictly less than 5τ.

BRIEF DESCRIPTION OF THE DRAWINGS

The invention will be better understood on reading the following description given purely by way of non-limiting example and with reference to the drawings in which:

FIG. 1 is a schematic illustration of an electric vehicle equipped with an RLC filter upstream of an electric converter,

FIG. 2 is a simplified equivalent circuit diagram of the RLC filter of the vehicle of FIG. 1,

FIGS. 3 and 4 are flow charts of deadbeat control methods for regulating the voltage U_(c) and the line current I_(l), respectively, of the RLC filter of the vehicle of FIG. 1,

FIG. 5 is a state diagram of a deadbeat control method for regulating both the voltage U_(c) and the line current I_(l) of the RLC filter of FIG. 1,

FIG. 6 is an illustration of a disturbance of the line voltage U_(l),

FIG. 7 is a graph illustrating the change over time in the voltage U_(c), the line current I_(l), and a current I_(u), in response to the disturbance represented on the graph of FIG. 6 in the absence of the methods of FIGS. 3 and 4, and

FIG. 8 is a graph representing the change over time of the same quantities as represented in FIG. 7 but for the case in which the regulation method implemented is that of FIG. 5.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT

In these drawings, the same references are used to denote the same items.

Hereafter in this description, features and functions that are well known to a person skilled in the art are not described in detail.

FIG. 1 represents an electric vehicle 2 equipped with a pantograph 4 sliding against an overhead catenary 6. The vehicle 2 is, for example, a railway vehicle such as a train. The catenary 6 is in this example powered by a DC supply via several substations arranged at regular intervals along the catenary 6. These substations are, for example, separated one from the other by distances greater than several kilometers.

To simplify FIG. 1, only one substation 8 has been represented. This substation 8 is connected to a three-phase electrical power distribution network 10. The substation 8 converts the three-phase voltage into a DC voltage which is then delivered on the catenary 6. Typically, the substation 8 comprises a circuit-breaker 12 able to electrically isolate the catenary 6 from the network 10 if the current in the catenary 6 exceeds a limit I_(lmax).

The vehicle 2 is equipped with an electric motor 16 intended to rotationally drive the drive wheels of the vehicle via a drive shaft 18.

Here, the motor 16 is a three-phase synchronous or asynchronous motor. This motor 16 is powered by an electric converter 20 intended to generate a three-phase voltage supplying the motor 16 from a DC voltage U_(c). The motor 16 operates in traction mode and, alternately, as a three-phase voltage generator, for example, when the vehicle 2 brakes.

The stator time constant τ of the motor 16 is between 4 ms and 100 ms.

Here, the converter 20 is formed by three arms connected in parallel between input points 22 and 24. Each arm comprises two controllable switches connected in series via a middle point. Each middle point is connected to a respective phase of the motor 16.

The points 22 and 24 are connected to the conductors 26 and 28, respectively, of a DC bus via an RLC filter 30.

The conductor 26 is connected electrically to the pantograph 4 via various items of equipment that are not represented, such as for example a circuit-breaker, a transformer and a rectifier bridge, so as to be supplied with a DC voltage via the catenary 6. The conductor 28 is electrically connected to a reference potential 32, via the rails of a railway track, or a second conductor which can be overhead or in the form of a ground-based rail against which current return contact shoes slide.

The filter 30 is a low-pass RLC filter, the natural period T_(f) of which is strictly greater than the time constant τ of the motor 16. Preferably, the natural period T_(f) of the filter 30 is greater than at least five or ten times the time constant τ of the motor 16 so that it can fulfil its filtering function.

The filter 30 comprises two input points 34 and 36 connected to the conductors 26 and 28 respectively so as to receive between these input points the line voltage U_(l). The filter 30 also comprises two output points 38 and 40 electrically connected to the input points 22 and 24, respectively, of the converter 20. A resistance R and an inductance L are connected in series between the points 34 and 38. The resistance R and the inductance L have been represented here as two separate items. However, in practice, the resistance R and the inductance L can be formed by one and the same component such as a winding.

The current flowing through the inductance L is denoted by I_(l). This current is called the line current.

The filter 30 also comprises a capacitor C electrically connected directly between the output points 38 and 40. The voltage across the terminals of this capacitor C is denoted by U_(c). The current flowing through the point 38 is denoted by I_(u).

The vehicle 2 can also comprise an electric braking rheostat 46 connected between the points 38 and 40 and the points 22 and 24. Typically, this rheostat 46 is intended to dissipate the electrical energy produced by the motor 16 when the latter operates as a generator and when the catenary 6 or the vehicle 2 is not in conditions for retrieving the braking energy. For example, the rheostat 46 is formed by a resistance R_(h) connected in series with a controllable switch 48 between the output points 38 and 40. The switch 48 is controllable so as to regulate the intensity of the current flowing through the resistance R_(h).

The vehicle 2 also comprises a unit 50 for driving the rheostat 46 and the converter 20 based on, in addition, measurements made at the filter 30. To this end, the unit 50 is connected to a memory 52 containing instructions to execute one of the methods of FIGS. 3 to 5. The memory 52 also contains the values of the various operating limits which will be described further in detail with reference to FIGS. 3 and 4.

The unit 50 is constructed from one or more electronic computers.

Here, the unit 50 is, for example, formed by a computer 54 intended to establish, based on measurements made at the filter 30, a setting Ī_(uc) for the average intensity Ī_(u) of the DC current I_(u) over an interval T.

The unit 50 also comprises a control unit 56 intended to control both the rheostat 46 and the converter 20 in order to arrive at the setting Ī_(uc). To this end, the unit 56 is connected to the rheostat 46 and to the converter 20. The unit 56 is also able to control the converter 20 as a function of a setting Γ_(c) for the torque to be supplied by the motor 16 in order to accelerate or brake the vehicle 2.

FIG. 2 represents a simplified electric circuit diagram of the filter 30 on which the various sign conventions for the line voltage U_(l), the line current I_(l), the current I_(u) and the voltage U_(c) are defined. In FIG. 2, sensors 60, 62 and 64 for the line voltage U_(l), the voltage U_(c) and the intensity of the line current I_(l), respectively, have been represented. These sensors 60, 62 and 64 are connected to the computer 54.

FIG. 3 represents a deadbeat control method for regulating the voltage U_(c), the method being implemented by the driving unit 50.

Initially, at a step 80, a sampling period T is chosen. Hereafter in this description, t_(i) denotes the instant at which the various measurements made by the sensors 60, 62 and 64 are sampled, and t_(i+1) the next sampling instant. These instants t_(i) and t_(i+1) are separated by the time interval T.

Here, the interval T is chosen to be sufficiently small in order that over a single interval T the intensity of the current I_(u) does not have the time to reach its asymptotic value, i.e. U_(c)/R_(m), where R_(m) is the stator resistance of the motor 16. This is because this asymptotic value can be up to a hundred times higher than an upper limit I_(umax) that is acceptable for the intensity of the current I_(u). To this end, the interval T is therefore chosen to be strictly less than 5τ. Preferably, the interval T is chosen to be less than

$\frac{\tau}{5}\mspace{14mu}{or}\mspace{14mu}{\frac{\tau}{10}.}$ Here, in the context of electric vehicles, the interval T is generally less than 20 ms.

It is also beneficial to choose an interval T which is not too small such that the intensity of the current I_(u) has the time to vary in a significant way between the instants t_(i) and t_(i+1). To this end, here, the interval T is chosen to be greater than 100 μs.

Next, at the instant t_(i), at a step 82, the voltage U_(c), the line voltage U_(l) and the line current I_(l) are measured. The results of these measurements are denoted by U_(ci), U_(li) and I_(li) respectively. At the step 82, it is also possible to calculate the value of the inductance L of the filter 30 if that value varies as a function of the current I_(l).

At a next step 84, a voltage setting U_(cc) for the voltage U_(c) is fixed. For example, the voltage setting U_(cc) is defined from the average value of the voltages U_(l) measured at the previous sampling instants. The average is produced over a period of time that is strictly greater than the natural period T_(f) of the filter 30 and preferably at least ten times greater than the natural period T_(f). For example, the voltage setting U_(cc) is determined using following relationship:

$\begin{matrix} {U_{cc} = {\frac{1}{N}{\sum\limits_{i = 1}^{N}\left( {U_{li} - {RI}_{li}} \right)}}} & (1) \end{matrix}$ where:

N is the number of intervals T taken into account to calculate the average,

the voltages U_(li) are the line voltages measured at the previous instants i,

the intensities I_(li) are the line current intensities I_(l) measured at the previous instants t_(i), and

R is the resistance of the filter 30.

Next, at a step 86, an estimate I_(lp) of the intensity of the current I_(l) which will be reached at the instant t_(i+1), if at the instant t_(i+1) the voltage U_(c) is equal to the voltage setting U_(cc), is constructed. For example, the estimate I_(lp) is constructed from the following relationship:

$\begin{matrix} {I_{1p} = \frac{{\mu_{1} \cdot \mu_{2} \cdot C \cdot \left( {a_{2} - a_{1}} \right) \cdot U_{cc}} - {\left( {{a_{1} \cdot \mu_{2} \cdot {\mathbb{e}}^{\mu_{2} \cdot T}} - {a_{2}{\mu_{1} \cdot {\mathbb{e}}^{\mu_{1} \cdot T}}}} \right) \cdot I_{li}} + {\mu_{1 \cdot}{\mu_{2} \cdot C \cdot \left\lbrack {{a_{1} \cdot a_{2} \cdot \left( {\mu_{1} - \mu_{2}} \right) \cdot {\overset{\_}{U}}_{1}} + {\left( {{a_{1}{\mathbb{e}}^{\mu_{2} \cdot T}} - {a_{2} \cdot {\mathbb{e}}^{\mu_{1} \cdot T}}} \right) \cdot U_{ci}}} \right\rbrack}}}{\left( {{a_{2} \cdot \mu_{1}} - {a_{1} \cdot \mu_{2}}} \right)}} & (2) \end{matrix}$ where:

C is the value of the capacitance C of the filter 30,

Ū_(l) is the average value of the voltage U_(l) over the interval T,

μ₁, μ₂, a₁, a₂ are defined below,

e^(x) is the exponential function.

It is assumed here that the line voltage is constant over the interval T such that the average value Ū_(l) is equal to U_(li).

μ₁ and μ₂ are the eigenvalues of an evolution matrix of the filter 30. These values are defined by the following relationships:

$\begin{matrix} {\mu_{1} = \frac{{{- R} \cdot C} + \sqrt{{R^{2} \cdot C^{2}} - {4 \cdot L \cdot C}}}{2 \cdot L \cdot C}} & (3) \\ {\mu_{2} = \frac{{{- R} \cdot C} - \sqrt{{R^{2} \cdot C^{2}} - {4 \cdot L \cdot C}}}{2 \cdot L \cdot C}} & (4) \end{matrix}$ where R, C and L are the values of the resistance R, the capacitance of the capacitor C and the value of the inductance L, respectively, of the filter 30.

a₁ and a₂ are defined by the following relationships:

$\begin{matrix} {a_{1} = \frac{{\mathbb{e}}^{\mu_{1} \cdot T} - 1}{\mu_{1}}} & (5) \\ {a_{2} = \frac{{\mathbb{e}}^{\mu_{2} \cdot T} - 1}{\mu_{2}}} & (6) \end{matrix}$

The relationship (2) has been obtained from the following discretized state equations of the filter 30, after elimination of Ī_(u) in the following system of equations (7) and (8):

$\begin{matrix} {{I_{{li} + 1} - {\mu_{2} \cdot C \cdot U_{{ci} + 1}}} = {{{\mathbb{e}}^{\mu_{1} \cdot T} \cdot \left( {I_{li} - {\mu_{2} \cdot C \cdot U_{ci}}} \right)} + {a_{1} \cdot \begin{pmatrix} {{\mu_{2} \cdot {\overset{\_}{I}}_{u}} +} \\ {\frac{1}{L} \cdot {\overset{\_}{U}}_{l}} \end{pmatrix}}}} & (7) \\ {{I_{{li} + 1} - {\mu_{1} \cdot C \cdot U_{{ci} + 1}}} = {{{\mathbb{e}}^{\mu_{2} \cdot T} \cdot \left( {I_{li} - {\mu_{1} \cdot C \cdot U_{ci}}} \right)} + {a_{2} \cdot \begin{pmatrix} {{\mu_{1} \cdot {\overset{\_}{I}}_{u}} +} \\ {\frac{1}{L} \cdot {\overset{\_}{U}}_{l}} \end{pmatrix}}}} & (8) \end{matrix}$ where:

U_(ci) and I_(li) are the voltage UC and the intensity of the current I_(l) measured at the instant t_(i+1).

I_(l,i+1) and U_(c,i+1) are the intensity of the current I_(l) and the voltage U_(c) at the instant t_(i+1).

Given that the regulation method implemented is a deadbeat control method, at the instant t_(i+1), the voltage U_(c,i+1) is equal to the voltage setting U_(cc). Furthermore, assuming that the line voltage U_(l) is constant over the interval T, the average voltage Ū_(l) is equal to U_(li). Under these conditions, the relationships (7) and (8) form a system of two equations with two unknowns, i.e. I_(l,i+1) and Ī_(u). It is therefore possible to solve this system of equations analytically in order to obtain the estimate I_(lp) (2) which corresponds to the value I_(l,i+1), eliminating Ī_(u) in the system of equations (7) and (8).

The way in which the relationships (7) and (8) have been obtained is described further in detail at the end of this description, in a section entitled “Establishing discretized state equations”.

Next, at a step 88, the estimate I_(lp) is compared with operating limits I_(lmin) and I_(lmax), within which the intensity of the line current I_(l) must be maintained. For example, the limit I_(lmin) is chosen to correspond to a saturated state of the inductance L, thereby providing for keeping the inductance L saturated as long as the method of FIG. 3 is executed. This exhibits the advantage of avoiding a scenario in which the inductance suddenly releases a high amount of energy when it desaturates. The limit I_(lmax) is that defined in relation to the circuit-breaker 12, or to the circuit-breaker on board the vehicle for protecting onboard equipment, according to their respective tripping limit by vehicle. For the circuit-breaker 12, this tripping limit is related to each vehicle according to the number of vehicles capable of circulating at a given time on the portion of line powered by the substation protected by the circuit-breaker 12, for example: tripping limit of the substation divided by the maximum number of vehicles.

If the estimate I_(lp) does not fall within the range [I_(lmin), I_(lmax)], at a step 90 the computer 54 modifies the voltage setting U_(cc) to obtain a temporary voltage setting U_(ccm) which provides for obtaining at the instant t_(i+1) an intensity of the current I_(l) contained within the range [I_(lmin), I_(lmax)]. For example, the temporary voltage setting U_(ccm) is obtained from the following relationship:

$\begin{matrix} {U_{ccm} = \frac{\begin{matrix} {{\left( {{a_{2} \cdot \mu_{1}} - {a_{1} \cdot \mu_{2}}} \right)I_{l\; m}} + {\left( {{a_{1} \cdot \mu_{2} \cdot {\mathbb{e}}^{\mu_{2} \cdot T}} - {a_{2} \cdot \mu_{1} \cdot {\mathbb{e}}^{\mu_{1} \cdot T}}} \right) \cdot}} \\ {I_{li} - {\mu_{1} \cdot \mu_{2} \cdot C \cdot \begin{bmatrix} {{a_{1} \cdot a_{2} \cdot \left( {\mu_{1} - \mu_{2}} \right) \cdot {\overset{\_}{U}}_{l}} +} \\ {\left( {{a_{1} \cdot {\mathbb{e}}^{\mu_{2} \cdot T}} - {a_{2} \cdot {\mathbb{e}}^{\mu_{1} \cdot T}}} \right) \cdot U_{ci}} \end{bmatrix}}} \end{matrix}}{\mu_{1} \cdot \mu_{2} \cdot C \cdot \left( {a_{2} - a_{1}} \right)}} & (9) \end{matrix}$ where I_(lm) is a limit value of the intensity of the line current I_(l) chosen from the set {I_(lmin); I_(lmax)}.

More specifically, I_(lm) is chosen to be equal to I_(lmin) if the estimate I_(lp) constructed at the step 86 is smaller than the limit I_(lmin). Otherwise, i.e. if this estimate is greater than the limit I_(lmax), then the value of the intensity I_(lm) is chosen to be equal to the limit I_(lmax).

The relationship (9) provides for obtaining the temporary voltage setting U_(ccm) that is the closest to the initial voltage setting U_(cc), while maintaining the intensity of the line current I_(l) within the range [I_(lmin), I_(lmax)].

After the step 90, at a step 92 the computer 54 calculates a current setting Ī_(uc) for the average intensity of the current I_(u) between the instants t_(i) and t_(i+1).

If at the step 88, the estimate I_(lp) is contained within the range [I_(lmin), I_(lmax)], then the computer 54 proceeds directly to the step 92.

The setting Ī_(uc) is determined analytically so that, at the instant t_(i+1), the voltage U_(c) is exactly equal to the voltage setting U_(cc) or to the temporary voltage setting U_(ccm) if the step 90 has been executed. For example, the current setting Ī_(uc) is established from the following relationship:

$\begin{matrix} {{\overset{\_}{I}}_{uc} = \frac{\begin{matrix} {{\left( {{\mathbb{e}}^{\mu_{1} \cdot T} - {\mathbb{e}}^{\mu_{2} \cdot T}} \right) \cdot I_{li}} + {C \cdot}} \\ \begin{bmatrix} {{\mu_{1} \cdot \mu_{2} \cdot \left( {a_{1} - a_{2}} \right) \cdot {\overset{\_}{U}}_{l}} - {\left( {\mu_{1} - \mu_{2}} \right) \cdot}} \\ {U_{cc} - {\left( {{\mu_{2} \cdot {\mathbb{e}}^{\mu_{1} \cdot T}} - {\mu_{1} \cdot {\mathbb{e}}^{\mu_{2} \cdot T}}} \right) \cdot U_{ci}}} \end{bmatrix} \end{matrix}}{\left( {{a_{2} \cdot \mu_{1}} - {a_{1\;} \cdot \mu_{2}}} \right)}} & (10) \end{matrix}$

The relationship (10) is obtained by solving the discretized state equations defined by the relationships (7) and (8) in order to extract from them the unknown Ī_(u), after elimination of I_(l,i+1) in the system of equations (7) and (8).

Next, at a step 94, the current setting Ī_(uc) is compared with an acceptable predetermined operating range [Ī_(u min), Ī_(u max)].

By way of example, the limit Ī_(u min) is chosen to be equal to the minimum average intensity of the current I_(u) that can be generated by the converter 20 when the motor 16 operates as a generator. This average is established over the sampling period T. The limit Ī_(u min) is negative since the motor is operating as a generator.

The limit Ī_(u max) is, for its part, for example, chosen to be equal to the sum of the maximum average intensity of the current that can be absorbed by the rheostat 46 and the maximum average intensity of the current that can be absorbed by the converter 20. The maximum average intensity that can be absorbed by the converter 20 is a function of the electrical characteristics of this converter. The maximum average intensity that can be absorbed by the rheostat is, for example, given by the ratio of a maximum allowable voltage U_(cmax) between the conductors 26 and 28 to the value of the resistance R_(h). These averages are established over the sampling period T.

If the current setting Ī_(uc) is in the range [Ī_(u min), Ī_(u max)] then the setting is transmitted as such to the control unit 56 at a step 96. Otherwise, one of the two limits Ī_(u min) or Ī_(u max) is transmitted to the unit 56 at a step 98.

More specifically, at the step 98, the limit Ī_(u min) is transmitted to the unit 56 as a current setting Ī_(uc) if the current setting Ī_(uc) established at the step 92 is less than the limit Ī_(u min). Otherwise, it is the limit Ī_(u max) which is transmitted as the setting of the average intensity of the current I_(u) to the unit 56.

After the step 96 or 98, at a step 100 the unit 56 controls the converter 20 and if necessary the rheostat 46 over the interval T in order to produce a current I_(u), the average intensity of which is equal to the current setting Ī_(uc). More specifically, for the case in which the current setting Ī_(uc) is positive, i.e. when current is being consumed, the unit 56 can control either only the rheostat 46 or only the converter 20, or both the rheostat 46 and the converter 20 to produce a current I_(u), the average intensity of which over the interval T is equal to the current setting Ī_(uc).

For example, if only the rheostat 46 is controlled, the duty factor or angle of opening of the switch 48 is calculated from the following relationship:

$\begin{matrix} {\alpha = \frac{R_{h} \cdot {\overset{\_}{I}}_{uc}}{{\overset{\_}{U}}_{c}}} & (11) \end{matrix}$ where Ū_(c) is the average value of the voltage U_(c) between the instant t_(i) and t_(i+1).

The average value Ū_(c) can, for example, be calculated from the following relationship:

$\begin{matrix} {\overset{\_}{U_{c}} = \frac{U_{cc} + U_{ci}}{2}} & (12) \end{matrix}$

The unit 56 can also modify the setting Γ_(c) for the torque that the motor 16 must produce in order that the average intensity Ī_(u) is equal to the current setting Ī_(uc).

A simultaneous control of the converter 20 and the rheostat 46 to produce a current I_(u), the average intensity Ī_(u) of which is equal to a current setting Ī_(uc), is also possible.

For the case in which the current setting Ī_(uc) is negative, i.e. when the converter 20 generates current, the unit 56 controls only the converter 20.

Variations in the torque setting Γ_(c) which are required to produce a current I_(u), the average intensity Ī_(u) of which is equal to the setting Ī_(uc), are produced sometimes in the positive direction, sometimes in the negative direction around a nominal point. Furthermore, the modified torque setting Γ_(c) lasts only for the interval T which is very small compared to the time to speed up the electric motor. Thus, the driver or passengers of the vehicle 2 do not feel these very fast changes in torque.

After the step 100, the method returns to the step 82. The steps 82 to 100 are therefore repeated at each sampling instant.

FIG. 4 represents a deadbeat control method for regulating the intensity of the line current I_(l); this method starts with a step 110 that is identical to the step 80. Next, at a step 112, the voltages U_(ci), U_(li) and the line current I_(li) are measured. This step 112 is, for example, identical to the step 82.

At a step 114, a line current setting I_(lc) for the intensity of the line current I_(l) is fixed. The setting I_(lc) is chosen to be in the range [I_(lmin), I_(lmax)]. For example, the setting I_(lc) is chosen to be equal to I_(lmin) or I_(lmax).

Next, at a step 116, an estimate U_(cp) of the voltage U_(c) at the instant t_(i+1), if the intensity of the line current I_(l) at this instant is equal to the line current setting I_(lc), is constructed. For example, this voltage estimate U_(cp) is constructed from the following relationship:

$\begin{matrix} {U_{cp} = \frac{\begin{matrix} {{\left( {{a_{1} \cdot \mu_{2}} - {a_{2} \cdot \mu_{1}}} \right) \cdot I_{l\; c}} - {\begin{pmatrix} {{a_{1} \cdot \mu_{2} \cdot {\mathbb{e}}^{\mu_{2} \cdot T}} -} \\ {a_{2} \cdot \mu_{1} \cdot {\mathbb{e}}^{\mu_{1} \cdot T}} \end{pmatrix} \cdot}} \\ {I_{l\; i} + {\mu_{1} \cdot \mu_{2} \cdot C \cdot \begin{bmatrix} {{a_{1} \cdot a_{2} \cdot \left( {\mu_{1} - \mu_{2}} \right) \cdot {\overset{\_}{U}}_{l}} -} \\ {\left( {{a_{2} \cdot {\mathbb{e}}^{\mu_{1} \cdot T}} - {a_{1} \cdot {\mathbb{e}}^{\mu_{2} \cdot T}}} \right) \cdot U_{ci}} \end{bmatrix}}} \end{matrix}}{\left( {a_{1} - a_{2}} \right) \cdot \mu_{1} \cdot \mu_{2} \cdot {C.}}} & (13) \end{matrix}$

The relationship (13) is obtained by solving the system of state equations defined by the relationships (7) and (8) for the case in which I_(l,i+1) is equal to I_(lc), eliminating I_(u) in the system of equations (7) and (8).

At a step 118, the voltage estimate U_(cp) is compared with an acceptable operating range [U_(cmin), U_(cmax)].

The limit U_(cmax) is, for example, equal to the acceptable maximum voltage between the conductors 26 and 28 and beyond which the rheostat 46 is operated to clip any voltage exceeding this dimensioning limit of the converter 20.

The limit U_(cmin) is for example chosen to be at the minimum of the acceptable voltage for operation at reduced speed, beyond which the capacitor must be recharged from the line.

At a step 120, if the voltage estimate U_(cp) does not fall within the range [U_(cmin), U_(cmax)], then the line current setting I_(lc) is modified to obtain a temporary line current setting I_(lcm) which provides for maintaining the voltage U_(c) within the range [U_(cmin), U_(cmax)] at the instant t_(i+1). For example, here, the temporary line current setting I_(lcm) is chosen in order that, at the instant t_(i+1), the voltage U_(c) is equal to the limit U_(cmin) or to the limit U_(cmax). For example, to this end, the temporary setting I_(lcm) is constructed from the following relationship:

$\begin{matrix} {I_{l\; c\; m} = \frac{\begin{matrix} {{\left( {{a_{1} \cdot \mu_{2} \cdot {\mathbb{e}}^{\mu_{2} \cdot T}} - {a_{2} \cdot \mu_{1} \cdot {\mathbb{e}}^{\mu_{1} \cdot T}}} \right) \cdot I_{li}} - {\mu_{1} \cdot \mu_{2} \cdot C \cdot}} \\ \begin{bmatrix} {{a_{1} \cdot a_{2\;} \cdot \left( {\mu_{1} - \mu_{2}} \right) \cdot \overset{\_}{U_{l}}} -} \\ {{\begin{pmatrix} {{a_{2} \cdot {\mathbb{e}}^{\mu_{1} \cdot T}} -} \\ {a_{1} \cdot {\mathbb{e}}^{\mu_{2} \cdot T}} \end{pmatrix} \cdot U_{ci}} - {\left( {a_{1} - a_{2}} \right) \cdot U_{c\; m}}} \end{bmatrix} \end{matrix}}{{a_{1} \cdot \mu_{2}} - {a_{2} \cdot \mu_{1}}}} & (14) \end{matrix}$ where U_(cm) is a value chosen from the set {U_(cmin); U_(cmax)}

More specifically, the value U_(cm) is chosen to be equal to U_(cmin) if the estimate U_(cp) is less than the limit U_(cmin). Otherwise, i.e. if the estimate U_(cp) is greater than the limit U_(cmax), the value U_(cm) is chosen to be equal to the limit U_(cmax).

After the step 120, or if the estimate U_(cp) falls within the range [U_(cmin), U_(cmax)], a step 122 for calculating a current setting Ī_(uc) for the average intensity of the current I_(u) between the instants t_(i) and t_(i+1) is carried out. More specifically, the current setting Ī_(uc) is calculated in order that, exactly at the instant t_(i+1), the intensity of the line current I_(l) is equal to the line current setting I_(lc) or to the temporary line current setting I_(lcm) if the step 120 has been executed.

For example, the setting Ī_(uc) is calculated from the following relationship:

$\begin{matrix} {{\overset{\_}{I}}_{uc} = \frac{\begin{matrix} {{\left( {\mu_{1} - \mu_{2}} \right) \cdot I_{l\; c}} - {\left( {{u_{1} \cdot {\mathbb{e}}^{\mu_{1} \cdot T}} - {\mu_{2} \cdot {\mathbb{e}}^{\mu_{2} \cdot T}}} \right) \cdot}} \\ {I_{l\; i} + {\mu_{1} \cdot \mu_{2} \cdot C \cdot \left( {{\mathbb{e}}^{\mu_{1} \cdot T} - {\mathbb{e}}^{\mu_{2} \cdot T}} \right) \cdot \left( {U_{ci} - {\overset{\_}{U}}_{l}} \right)}} \end{matrix}}{\mu_{1} \cdot \mu_{2} \cdot \left( {a_{1} - a_{2}} \right)}} & (15) \end{matrix}$

The relationship (15) is obtained by solving the system of discretized state equations defined by the relationships (7) and (8) for the case in which I_(l,i+1) is equal to I_(lc) and Ī_(u) is equal to Ī_(uc), after elimination of U_(c,i+1).

Next, the unit 56 executes steps 124, 126, 128, 130 which are identical to the steps 94, 96, 98 and 100, respectively, of the method of FIG. 3.

Given that the driving unit 50 cannot act on the line voltage U_(l) which is fixed by the voltage of the catenary 6, only the intensity of the current I_(u) can be controlled. Under these conditions, over an interval T, it is only possible to regulate either only the voltage U_(c) or only the line current I_(l). In other words, the methods of FIGS. 3 and 4 cannot be executed simultaneously. On the other hand, it is possible to regulate, alternately, the voltage U_(c) and the line current I_(l) by executing the methods of FIGS. 3 and 4 alternately. This has the effect, for example, of stabilizing the voltage U_(c) while maintaining the line current I_(l) within the operating range [I_(lmin), I_(lmax)]. For example, to prevent exceeding the limit I_(lmax), the method of FIG. 5 is implemented.

The method of FIG. 5 includes:

-   -   a phase 140 for regulating the voltage U_(c) using the method of         FIG. 3, and     -   a phase 142 for regulating the line current I_(l) using the         method of FIG. 4.

The unit 50 switches automatically from the phase 140 to the phase 142 when the measured intensity I_(li) becomes strictly greater than the limit I_(lmax).

Conversely, the unit 50 switches automatically from the phase 142 to the phase 140 when the intensity I_(li) becomes less than the predetermined limit, for example, equal to I_(lmax).

Thus, as long as the measured intensity of the current I_(l) is strictly less than the limit I_(lmax), the voltage U_(c) is kept equal at each instant to the voltage setting U_(cc). Oscillations of the setting voltage U_(cc) following a disturbance of the line voltage U_(l) are therefore limited. If the intensity of the line current I_(l) exceeds the limit I_(lmax), then the first objective involving regulating the voltage U_(c) is abandoned and a switchover to the phase 142 takes place. For example, at the phase 142, the line current setting I_(lc) can be chosen to be equal to the limit I_(lmax) or strictly less than the limit I_(lmax). The phase 142 stops as soon as the measured intensity of the line current I_(l) is less than or equal to the limit I_(lmax) and then a switch back to the phase 140 takes place.

Thus, by alternating the phases 140 and 142 in time, the voltage U_(c) can be stabilized while maintaining the intensity of the line current I_(l) less than the limit I_(lmax).

Similarly, the phases 140 and 142 are executed alternately in order to maintain the intensity of the current I_(l) greater than the limit I_(lmin).

Operation of the method of FIG. 5 is illustrated for the particular case of a disturbance of the line voltage U_(l) represented on the graph of FIG. 6. This disturbance involves making the line voltage U_(l) drop instantaneously from 3000 volts to 2400 volts and maintaining the line voltage U_(l) equal to 2400 volts for 0.1 seconds. Next, the line voltage U_(l) rises instantaneously to 3450 volts and remains equal to this value for 0.1 seconds before returning, instantaneously, to 3000 volts. Variations in the line voltage U_(l) take place instantaneously in this case. It is therefore understood that what is represented in FIG. 6 is only a theoretical disturbance.

The graphs represented in FIGS. 7 and 8 have been obtained from a simulation of a model of the vehicle 2 with the following numeric values:

-   -   L=3 mH,     -   R=25 mΩ,     -   C=18 mF,     -   U_(l)=3000 V,     -   U_(cmax)=3500 V,     -   I_(umax)=470 A,     -   I_(umin)=−470 A,     -   I_(lmax)=2000 A,     -   R_(h)=2Ω.

FIG. 7 represents the change as a function of time of the voltage U_(c), of the intensity of the line current I_(l), and of the intensity of the current I_(u), for the case in which no method for regulating the voltage U_(c) or the line current I_(l) is implemented. As can be observed, this results in strong oscillations of the voltage U_(c) and of the intensity of the line current I_(l).

With everything else being equal, the graph of FIG. 8 represents the change over time of the voltage U_(c), and of the intensities of the currents I_(l) and I_(u), in response to the disturbance represented in FIG. 6 when the method of FIG. 5 is implemented. Furthermore, the voltage setting U_(cc) is chosen here to be equal to the average of the line voltage U_(l) over the last ten milliseconds so as to impose sudden variations of this setting which necessarily result in the creation of situations in which the current setting Ī_(uc) reaches the limits of the range [Ī_(u min), Ī_(u max)]. It will be noted that this choice is made, in this case, only by way of illustration to show what happens when the setting Ī_(uc) reaches one of the limits Ī_(u min) or Ī_(u max). In practice, the voltage setting U_(cc) will be chosen in such a way as to smooth out the disturbances of the line voltage U_(l) as indicated with reference to the step 82.

As illustrated on the graph of FIG. 8, the voltage U_(c) is maintained close to the voltage setting U_(cc). It is therefore understood that using this method, variations in the voltage U_(c) are very well controlled even in the event of sudden variations of the line voltage U_(l).

Furthermore, as illustrated by the plateaus in the graph representing the change in the current I_(u) as a function of time, the limits Ī_(u max) and Ī_(u min) are reached so that during these plateaus, the voltage U_(c) is not strictly equal to the voltage setting U_(cc). On the other hand, outside these plateaus, the voltage U_(c) is equal to the voltage setting U_(cc).

Thus, as FIG. 8 illustrates, by virtue of the method of FIG. 5, variations in the voltage U_(c) are controlled while maintaining the intensities I_(u) and I_(l) within their respective operating ranges.

Many other embodiments are possible.

For example, the conductor 26 of the DC bus can be connected to the pantograph 4 via a rectifier such as a diode bridge rectifier and via a transformer in the case of a catenary supplied by an alternating single-phase voltage.

One from among the line current intensity I_(l) and the voltage U_(c) can be estimated instead of being measured. The intensity of the line current I_(l) and the voltage U_(c) can also both be estimated.

The above-described applies also to the case of DC motors. In that case, the converter 20 is, for example, a chopper/downconverter.

Lastly, it will be noted that technical constraints can require the use of an approximation Î_(uc) of the current setting Ī_(uc) and not the exact value given by the relationship (10) or (15). For example, one of these technical constraints is the number of digits after the decimal point that the computer 54 can generate. Thus, in this description, it is considered that from a practical point of view, a current setting Î_(uc) is established from the system of state equations defined by the relationships (7) and (8) if the following intercorrelation coefficient α is greater than 0.9:

$\begin{matrix} {\alpha = {\frac{1}{NT}{\int_{0}^{NT}{\frac{{{\overset{\_}{I}}_{uc}(t)}{{\hat{I}}_{uc}(t)}}{\sqrt{{\overset{\_}{I}}_{uceff}{\hat{I}}_{uceff}}}\ {\mathbb{d}t}}}}} & (16) \end{matrix}$ where:

N is a whole number, greater than 20, of intervals T taken into account to calculate the intercorrelation coefficient α,

Ī_(uc)(t) is the exact value of the setting for the average intensity of the current I_(u), obtained from the relationship (10) or (15),

Î_(uc)(t) is the approximation of the setting Ī_(uc)(t) sent by the computer 54 to the unit 56,

Ī_(uceff) is defined by the following relationship:

$\begin{matrix} {{\overset{\_}{I}}_{uceff} = {\frac{1}{NT}{\int_{0}^{NT}{{{\overset{\_}{I}}_{uc}^{2}(t)}\ {\mathbb{d}t}}}}} & (17) \end{matrix}$

Î_(uceff) is defined by the following relationship:

$\begin{matrix} {{\hat{I}}_{uceff} = {\frac{1}{NT}{\int_{0}^{NT}{{{\hat{I}}_{uc}^{2}(t)}{\mathbb{d}t}}}}} & (18) \end{matrix}$

As defined above, the intercorrelation coefficient α represents the degree of correlation between the approximation Î_(uc) and the exact current setting Ī_(uc).

Preferably, if approximations must be made, they will be produced in such a way that the intercorrelation coefficient α defined above is even greater than 0.7 or 0.99.

Establishing Discretized State Equations for the Filter 30 I Electrotechnical Model of the Line Filter

I-1—System of Differential Equations for the RLC Filter

${U_{l} - U_{c}} = {{L \cdot \frac{\mathbb{d}I_{l}}{\mathbb{d}t}} + {R \cdot I_{l}}}$ ${I_{l} - I_{u}} = {C \cdot \frac{\mathbb{d}U_{c}}{\mathbb{d}t}}$ I-2—System of State Equations

The filter is a second-order system, and it therefore has two degrees of freedom. The state vector is therefore a vector with two dimensions. The line current and the voltage of the capacitor can be chosen to be the two coordinates of the state vector of the filter:

${\overset{\rightarrow}{X}(t)} = \begin{bmatrix} {I_{l}(t)} \\ {U_{c}(t)} \end{bmatrix}$

These two variables are measured and are therefore known. If this were not the case, it would be necessary to estimate one of them or observe it.

The control variables of the system are the usage current and the line voltage. Since the line voltage cannot be modified directly, it features in the model as a control variable that is measured and not calculated.

$\overset{\rightarrow}{V} = \begin{bmatrix} I_{u} \\ U_{l} \end{bmatrix}$

The system of state equations thus defined can be expressed from the following differential equations:

$\frac{\mathbb{d}I_{l}}{\mathbb{d}t} = {{{- \frac{R}{L}} \cdot I_{l}} - {\frac{1}{L} \cdot U_{c}} + {\frac{1}{L} \cdot U_{l}}}$ $\frac{\mathbb{d}U_{c}}{\mathbb{d}t} = {{\frac{1}{C} \cdot I_{l}} - {\frac{1}{C} \cdot I_{u}}}$

The continuous-time state equation is hence deduced:

$\begin{bmatrix} \frac{\mathbb{d}I_{l}}{\mathbb{d}\; t} \\ \frac{\mathbb{d}U_{c}}{\mathbb{d}t} \end{bmatrix} = {{\begin{bmatrix} {- \frac{R}{L}} & {- \frac{1}{L}} \\ \frac{1}{C} & 0 \end{bmatrix} \cdot \begin{bmatrix} I_{l} \\ U_{c} \end{bmatrix}} + {\begin{bmatrix} 0 & \frac{1}{L} \\ {- \frac{1}{C}} & 0 \end{bmatrix} \cdot \begin{bmatrix} I_{u} \\ U_{l} \end{bmatrix}}}$

By comparing with the system of equations of the continuous-time form:

$\overset{.}{\overset{\rightarrow}{X}} = {{A \cdot \overset{\rightarrow}{X}} + {B \cdot \overset{\rightarrow}{V}}}$ $\overset{\rightarrow}{Y} = {E \cdot \overset{\rightarrow}{X}}$ this  gives:   $\overset{.}{\overset{\rightarrow}{X}} = {{\begin{bmatrix} {- \frac{R}{L}} & {- \frac{1}{L}} \\ \frac{1}{C} & 0 \end{bmatrix} \cdot \overset{\rightarrow}{X}} + {\begin{bmatrix} 0 & \frac{1}{L} \\ {- \frac{1}{C}} & 0 \end{bmatrix} \cdot \overset{\rightarrow}{V}}}$ $\overset{\rightarrow}{Y} = {E \cdot \overset{\rightarrow}{X}}$

Furthermore, given that the measurement vector is the state vector:

$A = {{\begin{bmatrix} {- \frac{R}{L}} & {- \frac{1}{L}} \\ \frac{1}{C} & 0 \end{bmatrix}\; B} = {{\begin{bmatrix} 0 & \frac{1}{L} \\ {- \frac{1}{C}} & 0 \end{bmatrix}\; E} = I_{2}}}$ where:

I₂ is the unit matrix of dimension 2,

A is the free evolution matrix, and

B is the control matrix.

II Diagonalization of the Evolution Matrix

II-1—Characteristic Equation of the Evolution Matrix

The determinant of the matrix is: μ·I−A, equal to 0: therefore:

${\det\begin{bmatrix} {\mu + \frac{R}{L}} & \frac{1}{L} \\ {- \frac{1}{C}} & \mu \end{bmatrix}} = 0$ ${{\mu \cdot \left( {\mu + \frac{R}{L}} \right)} + \frac{1}{L \cdot C}} = 0$ ${\mu^{2} + {\mu \cdot \frac{R}{L}} + \frac{1}{L \cdot C}} = 0$ II-2—Eigenvalues of the Evolution Matrix

These are the roots of the characteristic equation:

$\mu_{1} = \frac{{{- R} \cdot C} + \sqrt{{R^{2} \cdot C^{2}} - {4 \cdot L \cdot C}}}{2 \cdot L \cdot C}$ $\mu_{2} = \frac{{{- R} \cdot C} - \sqrt{{R^{2} \cdot C^{2}} - {4 \cdot L \cdot C}}}{2 \cdot L \cdot C}$

It is interesting to note that these eigenvalues are constant insofar as the inductance does not vary with the current, and that they are dependent on the current otherwise.

These roots verify the characteristic equation:

${\mu_{1} \cdot \left( {\mu_{1} + \frac{R}{L}} \right)} = {{\mu_{2} \cdot \left( {\mu_{2} + \frac{R}{L}} \right)} = {- \frac{1}{L \cdot C}}}$ which can be rewritten thus:

${{\left( {\mu_{i} + \frac{R}{L}} \right) \cdot \mu_{i}} + \frac{1}{L \cdot C}} = 0$

The characteristic equation also provides the sum and product of the eigenvalues as notable relationships:

${\mu_{1} + \mu_{2}} = {- \frac{R}{L}}$ ${\mu_{1} \cdot \mu_{2}} = \frac{1}{L \cdot C}$

It can also be noted that the double root μ₁=μ₂ is obtained for:

$R = {2 \cdot \sqrt{\frac{L}{C}}}$ which is the critical damping resistance.

If R=0, μ_(i)=±i·ω, where:

$\omega = {\frac{1}{\sqrt{L \cdot C}}.}$ The complex conjugate poles are recognized. If R≠0, the poles are complex conjugates with a negative real part and if the resistance is large enough, above the critical damping value, the poles are real and negative. II-3—Eigenvectors of the Evolution Matrix

These are calculated by:

${\left( {{\mu_{i} \cdot I} - A} \right) \cdot \prod\limits_{i}}\; = {{{0\begin{bmatrix} {\mu_{i} + \frac{R}{L}} & \frac{1}{L} \\ {- \frac{1}{C}} & \mu_{i} \end{bmatrix}} \cdot \begin{bmatrix} \pi_{1\; i} \\ \pi_{2\; i} \end{bmatrix}} = 0}$ i.e.:

The product of the matrices is then expressed as a system of equations:

${{\left( {\mu_{i} + \frac{R}{L}} \right) \cdot \pi_{1\; i}} + {\frac{1}{L} \cdot \pi_{2\; i}}} = {{0 - {\frac{1}{C} \cdot \pi_{1\; i}} + {\mu_{i} \cdot \pi_{2\; i}}} = 0}$

From the second equation of the system, the following is deduced: π_(1i)=μ_(i) ·C·π _(2i) and this relationship enables the first equation to be rewritten thus:

${\left\lbrack {{\left( {\mu_{i} + \frac{R}{L}} \right) \cdot \mu_{i}} + \frac{1}{L \cdot C}} \right\rbrack \cdot \pi_{2\; i}} = 0$

This equation is always verified for the two eigenvalues, regardless of the value of π_(2i)≠0, due to the fact that the first factor is in an identical manner zero according to the characteristic equation.

Therefore let π₂₁=−1 and π₂₂=1. The following is hence deduced: π_(1i)=μ_(i) ·C and π₁₂=μ₂ ·C.

The transformation matrix is formed by the eigenvectors:

${P = \begin{bmatrix} \pi_{11} & \pi_{12} \\ \pi_{21} & \pi_{22} \end{bmatrix}}\mspace{14mu}$ ${{{i.e.}:P} = \begin{bmatrix} {{- \mu_{1}} \cdot C} & {\mu_{2} \cdot C} \\ {- 1} & 1 \end{bmatrix}}\mspace{11mu}$

The inverse of the transformation matrix is:

$P^{- 1} = {\frac{1}{\left( {\mu_{2} - \mu_{1}} \right) \cdot C} \cdot \begin{bmatrix} 1 & {{- \mu_{2}} \cdot C} \\ 1 & {{- \mu_{1}} \cdot C} \end{bmatrix}}$ II-4—Diagonal Matrix

We can now write:

$\begin{matrix} {A = {{P \cdot D^{- 1} \cdot P^{- 1}}\mspace{14mu}{where}\text{:}}} \\ {D = \begin{bmatrix} \mu_{1} & 0 \\ 0 & \mu_{2} \end{bmatrix}} \\ {D^{- 1} = \begin{bmatrix} \frac{1}{\mu_{1}} & 0 \\ 0 & \frac{1}{\mu_{2}} \end{bmatrix}} \end{matrix}$

III Projection of the State Equations

III-1—Discretized State Equations

The discretized state equations are obtained by integration from the initial instant t_(i), until the end t_(i+1) of the sampling period of duration T: {right arrow over (X)} _(t) _(i+1) =F·{right arrow over (X)} _(t) _(i) +G·{right arrow over ( V _(t) _(i) _(→t) _(i+1) with: F=e ^(A·T) G=A ⁻¹·(e ^(A·T) −I)·B

If {right arrow over (X)}_(t) _(i) represents the state vector at the initial instant, {right arrow over (X)}_(t) _(i+1) then represents the prediction of the state vector. It will now be noted: {right arrow over (X)}_(i)={right arrow over (X)}_(t) _(i) and {right arrow over (X)} _(p) ={right arrow over (X)} _(t) _(i+1)

The transformation matrix P and the diagonal matrix of the evolution matrix are used to calculate the transition and control matrices: F=P·e ^(D·T) ·P ⁻¹ G=A ⁻¹ ·P·(e ^(D·T) −I)P ⁻¹ ·B

The system of discretized state equations can then be written: {right arrow over (X)} _(p) =P·e ^(D·T) ·P ⁻¹ · X _(i) +A ⁻¹ ·P·(e ^(D·T) −I)P ⁻¹ ·B·{right arrow over ( V III-2—Projection of the State Vectors

Now it is merely a case of projecting the system of discretized state equations in the base of eigenvectors by premultiplying by the inverse of the transformation matrix, and of isolating the state “eigenvectors”. P ⁻¹ ·{right arrow over (X)} _(p) =e ^(D·T) ·P ⁻¹ ·{right arrow over (X)} _(i) +A ⁻¹ ·P·(e ^(D·T) −I)·P ⁻¹ ·B·{right arrow over ( V noting that: P ⁻¹ ·A ⁻¹ ·P=(P ⁻¹ ·A·P)⁻¹ =D ⁻¹ the vector relationship is simplified: [P ⁻¹ ·{right arrow over (X)} _(p) ]=e ^(D·T) ·[P ⁻¹ ·{right arrow over (X)} _(i) ]+D ⁻¹·(e ^(D·T) −I)·[P ⁻¹ ·B]·{right arrow over ( V

To simplify the final representation of the state equation, let us multiply the two members of the equation by the constant: (μ₂−μ₁)·C: (μ₂−μ₁)·C·[P ⁻¹ ·{right arrow over (X)} _(P) ]=e ^(D·T)·(μ₂−μ₁)·C·[P ⁻¹ ·{right arrow over (X)} _(i) ]+D ⁻¹·(e ^(D·T) −I)·(μ₂−μ₁)·C·[P ⁻¹ ·B]·{right arrow over ( V The state eigenvectors are defined by:

$\begin{matrix} {\overset{->}{\Psi} = {\left( {\mu_{2} - \mu_{1}} \right) \cdot C \cdot \left\lbrack {P^{- 1} \cdot \overset{->}{X}} \right\rbrack}} \\ {\overset{->}{\Psi} = {\begin{bmatrix} 1 & {{- \mu_{2}} \cdot C} \\ 1 & {{- \mu_{1}} \cdot C} \end{bmatrix} \cdot \overset{->}{X}}} \\ {\overset{->}{\Psi} = {\begin{bmatrix} 1 & {{- \mu_{2}} \cdot C} \\ 1 & {{- \mu_{1}} \cdot C} \end{bmatrix} \cdot \begin{bmatrix} I_{l} \\ U_{c} \end{bmatrix}}} \end{matrix}$

Their coordinates are therefore:

$\begin{bmatrix} \Psi_{1} \\ \Psi_{2} \end{bmatrix} = \begin{bmatrix} {I_{l} - {\mu_{2} \cdot C \cdot U_{c}}} \\ {I_{l} - {\mu_{1} \cdot C \cdot U_{c}}} \end{bmatrix}$

The state matrix-equation can be expressed using this new definition, noting, moreover, that:

$\begin{matrix} {{\mathbb{e}}^{D \cdot T} = \begin{bmatrix} {\mathbb{e}}^{\mu_{1} \cdot T} & 0 \\ 0 & {\mathbb{e}}^{\mu_{2} \cdot T} \end{bmatrix}} \\ {{\overset{->}{\Psi}}_{p} = {{{\mathbb{e}}^{D \cdot T} \cdot {\overset{->}{\Psi}}_{i}} + {\begin{bmatrix} \frac{1}{\mu_{1}} & 0 \\ 0 & \frac{1}{\mu_{2}} \end{bmatrix} \cdot \begin{bmatrix} {{\mathbb{e}}^{\mu_{1} \cdot T} - 1} & 0 \\ 0 & {{\mathbb{e}}^{\mu_{2} \cdot T} - 1} \end{bmatrix} \cdot \begin{bmatrix} 1 & {{- \mu_{2}} \cdot C} \\ 1 & {{- \mu_{1}} \cdot C} \end{bmatrix} \cdot}}} \\ {\begin{bmatrix} 0 & \frac{1}{L} \\ {- \frac{1}{C}} & 0 \end{bmatrix} \cdot \begin{bmatrix} {\overset{\_}{I}}_{u} \\ {\overset{\_}{U}}_{l} \end{bmatrix}} \\ {{\overset{->}{\Psi}}_{p} = {{{\mathbb{e}}^{D \cdot T} \cdot {\overset{->}{\Psi}}_{i}} + {\begin{bmatrix} \frac{{\mathbb{e}}^{\mu_{1} \cdot T} - 1}{\mu_{1}} & 0 \\ 0 & \frac{{\mathbb{e}}^{\mu_{2} \cdot T} - 1}{\mu_{2}} \end{bmatrix} \cdot \begin{bmatrix} \mu_{2} & \frac{1}{L} \\ \mu_{1} & \frac{1}{L} \end{bmatrix} \cdot \begin{bmatrix} {\overset{\_}{I}}_{u} \\ {\overset{\_}{U}}_{l} \end{bmatrix}}}} \\ {{\overset{->}{\Psi}}_{p} = {{{\mathbb{e}}^{D \cdot T} \cdot {\overset{->}{\Psi}}_{i}} + {\begin{bmatrix} \frac{{\mathbb{e}}^{\mu_{1} \cdot T} - 1}{\mu_{1}} & 0 \\ 0 & \frac{{\mathbb{e}}^{\mu_{2} \cdot T} - 1}{\mu_{2}} \end{bmatrix} \cdot \begin{bmatrix} {{\mu_{2} \cdot {\overset{\_}{I}}_{u}} + {\frac{1}{L} \cdot {\overset{\_}{U}}_{l}}} \\ {{\mu_{1} \cdot {\overset{\_}{I}}_{u}} + {\frac{1}{L} \cdot {\overset{\_}{U}}_{l}}} \end{bmatrix}}}} \end{matrix}$

The control “eigenvector”{right arrow over (Ξ)} is defined by:

$\overset{->}{\Xi} = \begin{bmatrix} {{\mu_{2} \cdot {\overset{\_}{I}}_{u}} + {\frac{1}{L} \cdot {\overset{\_}{U}}_{l}}} \\ {{\mu_{1} \cdot {\overset{\_}{I}}_{u}} + {\frac{1}{L} \cdot {\overset{\_}{U}}_{l}}} \end{bmatrix}$

The system of state equations can now be written in a simplified manner:

$\begin{matrix} {{\overset{->}{\Psi}}_{p} = {{{\mathbb{e}}^{D \cdot T} \cdot {\overset{->}{\Psi}}_{i}} + {\begin{bmatrix} \frac{{\mathbb{e}}^{\mu_{1} \cdot T} - 1}{\mu_{1}} & 0 \\ 0 & \frac{{\mathbb{e}}^{\mu_{2} \cdot T} - 1}{\mu_{2}} \end{bmatrix} \cdot \overset{->}{\Xi}}}} \\ {\Psi_{1p} = {{{\mathbb{e}}^{\mu_{1} \cdot T} \cdot \Psi_{1i}} + {\frac{{\mathbb{e}}^{\mu_{1} \cdot T} - 1}{\mu_{1}} \cdot \Xi_{1}}}} \\ {\Psi_{2p} = {{{\mathbb{e}}^{\mu_{2} \cdot T} \cdot \Psi_{2i}} + {\frac{{\mathbb{e}}^{\mu_{2} \cdot T} - 1}{\mu_{2}} \cdot \Xi_{2}}}} \end{matrix}$ or:

Lastly, by letting:

$\begin{matrix} {a_{1} = \frac{{\mathbb{e}}^{\mu_{1} \cdot T} - 1}{\mu_{1}}} \\ {a_{2} = \frac{{\mathbb{e}}^{\mu_{2} \cdot T} - 1}{\mu_{2}}} \end{matrix}$ the “characteristic” state equations become:

$\begin{matrix} {\Psi_{1p} = {{{\mathbb{e}}^{\mu_{1} \cdot T} \cdot \Psi_{1i}} + {a_{1} \cdot \Xi_{1}}}} \\ {\Psi_{2p} = {{{\mathbb{e}}^{\mu_{2} \cdot T} \cdot \Psi_{2i}} + {a_{2} \cdot \Xi_{2}}}} \end{matrix}$

The previous system can now be explained using the definition of intermediate variables:

$\begin{matrix} {{I_{lp} - {\mu_{2} \cdot C \cdot U_{cp}}} = {{{\mathbb{e}}^{\mu_{1} \cdot T} \cdot \left( {I_{li} - {\mu_{2} \cdot C \cdot U_{ci}}} \right)} + {a_{1} \cdot \left( {{\mu_{2} \cdot {\overset{\_}{I}}_{u}} + {\frac{1}{L} \cdot {\overset{\_}{U}}_{l}}} \right)}}} \\ {{I_{lp} - {\mu_{1} \cdot C \cdot U_{cp}}} = {{{\mathbb{e}}^{\mu_{2} \cdot T} \cdot \left( {I_{li} - {\mu_{1} \cdot C \cdot U_{ci}}} \right)} + {a_{2} \cdot \left( {{\mu_{1} \cdot {\overset{\_}{I}}_{u}} + {\frac{1}{L} \cdot {\overset{\_}{U}}_{l}}} \right)}}} \end{matrix}$ 

1. A method for regulating a voltage U_(c) between a first and a second output point of a low-pass RLC filter of natural period T_(f), the RLC filter including two input points electrically connected, respectively, to conductors of a DC bus of an electric vehicle powered via a catenary, first and second output points being electrically connected to a controllable electric converter for controlling torque exerted by an electric traction motor of the electric vehicle, stator time constant τ of the electric traction motor being strictly less than the natural period T_(f), the method including measuring or estimating the intensity I_(li) of a line current I_(l) flowing through an inductance of the filter at an instant t_(i), of a voltage U_(ci) between the first and second output points of the filter at the instant t_(i), and of a line voltage U_(l) between the input points of the filter, wherein the method is a deadbeat control method comprising: calculating a setting Ī_(uc) for an average intensity Ī_(u) of a DC current I_(u) flowing through the first output point of the filter between the instant t_(i) and an instant t_(i+1), the setting Ī_(uc) being established from discretized state equations of the filter in such a way that the voltage U_(c) is equal to a predetermined voltage setting U_(cc) at the instant t_(i+1), the discretized state equations between them establishing relationships between the intensities and I_(li) to I_(l,i+1) of a line current I_(l) at the instants t_(i) and t_(i+1) respectively, the voltages t_(ci) and U_(c,i+1) between the first and second output points of the filter at the instants t_(i) and t_(i+1) respectively, the average line voltage Ū_(l) between the instants t_(i) and t_(i+1) and the average intensity Ī_(u); and controlling the electric converter in order to produce a current I_(u) flowing through the first output point of the filter, the average intensity Ī_(u) of which between the instants t_(i) and t_(i+1) is equal to the setting Ī_(uc), the time interval T between the instants t_(i) and t_(i+1) being strictly less than 5τ.
 2. The method according to claim 1, wherein the voltage setting U_(cc) is chosen always to be less than or equal to a limit U_(cmax), the limit U_(cmax) corresponding to the maximum allowable voltage at the input of the electric converter or between the output points of the filter.
 3. The method according to claim 1, wherein the method includes: constructing an estimate I_(lp) of the line current I_(l) which will be reached if the voltage U_(c) is equal to the voltage setting U_(cc) at the instant t_(i+1); comparing the estimate I_(lp) with at least one predetermined limit I_(lm); only if the predetermined limit I_(lm) is crossed, modifying the voltage setting U_(cc) so as to obtain a temporary voltage setting U_(ccm) which corresponds to an estimate I_(lp) that does not cross the predetermined limit I_(lm), and using the temporary voltage setting U_(ccm) in place of the voltage setting U_(cc) during the calculation of the setting Ī_(uc) only for the interval T in progress; and if the predetermined limit I_(lm) is not crossed, using the voltage setting U_(cc) for calculating the setting Ī_(uc) for the interval T in progress.
 4. The method according to claim 1, wherein the setting Ī_(uc) is a solution of the following system of equations: $\begin{matrix} {{I_{lp} - {\mu_{2} \cdot C \cdot U_{cc}}} = {{{\mathbb{e}}^{\mu_{1} \cdot T} \cdot \left( {I_{li} - {\mu_{2} \cdot C \cdot U_{ci}}} \right)} + {a_{1} \cdot \left( {{\mu_{2} \cdot {\overset{\_}{I}}_{uc}} + {\frac{1}{L} \cdot {\overset{\_}{U}}_{l}}} \right)}}} \\ {{I_{lp} - {\mu_{1} \cdot C \cdot U_{cc}}} = {{{\mathbb{e}}^{\mu_{2} \cdot T} \cdot \left( {I_{li} - {\mu_{1} \cdot C \cdot U_{ci}}} \right)} + {a_{2} \cdot \left( {{\mu_{1} \cdot {\overset{\_}{I}}_{uc}} + {\frac{1}{L} \cdot {\overset{\_}{U}}_{l}}} \right)}}} \end{matrix}$ where: R and L are the values of the resistance and the inductance, respectively, of the RLC filter, and which are connected in series between the first input and output points; C is the capacitance of the capacitor connected between the first and second output points; μ₁ and μ₂ are the eigenvalues of an evolution matrix and are defined by the following relationships: $\begin{matrix} {\mu_{1} = \frac{{{- R} \cdot C} + \sqrt{{R^{2} \cdot C^{2}} - {4 \cdot L \cdot C}}}{2 \cdot L \cdot C}} \\ {{\mu_{2} = \frac{{{- R} \cdot C} - \sqrt{{R^{2} \cdot C^{2}} - {4 \cdot L \cdot C}}}{2 \cdot L \cdot C}};{and}} \end{matrix}$ a₁ and a₂ are values defined by the following relationships: $\begin{matrix} {a_{1} = \frac{{\mathbb{e}}^{\mu_{1} \cdot T} - 1}{\mu_{1}}} \\ {a_{2} = {\frac{{\mathbb{e}}^{\mu_{2} \cdot T} - 1}{\mu_{2}}.}} \end{matrix}$
 5. The method according to claim 1, wherein the voltage setting U_(cc) is constructed from the line voltage U_(l) in such a way that its power spectrum does not exhibit any harmonic beyond the frequency 0.9/T_(f).
 6. The method according to claim 1, wherein the method includes controlling a rheostat in order to produce, in combination with the control of the converter, the current I_(u) flowing through the first output terminal of the filter, the average intensity Ī_(u) of which between the instants t_(i) and t_(i+1) is equal to the current setting Ī_(uc).
 7. The method according to claim 1, wherein the interval T is less than or equal to τ/5.
 8. An information recording medium comprising instructions for executing a method as recited in claim 1 when these instructions are executed by an electronic computer.
 9. A method for regulating a line current I_(l) flowing through an inductance L of a low-pass RLC filter of natural period T_(f), this filter including: two input points electrically connected, respectively, to the conductors of a DC bus of an electric vehicle powered via a catenary; and first and second output points, the first output point being electrically connected to a controllable electric converter in order to cause the torque of an electric traction motor of the electric vehicle to vary, the stator time constant τ of this electric motor being strictly less than the natural period T_(f), the method including: measuring or estimating of the intensity I_(li) of the line current I_(l) at an instant t_(i), of a voltage U_(ci) between the output points of the filter at the instant t_(i), and of a line voltage U_(l) between the input points of the filter, wherein the method is a deadbeat control method including: calculating a setting Ī_(uc) for the average intensity Ī_(u) of a DC current I_(u) flowing through the first output point of the filter between the instant t_(i) and an instant t_(i+1), the setting Ī_(uc) being established from discretized state equations of the filter in such a way that the intensity of the line current I_(l) is equal to a predetermined line current setting T_(lc) at the instant t_(i+1), the discretized state equations between them establishing relationships between the intensities I_(li) and I_(l,i+1) of the line current I_(l) at the instants t_(i) and t_(i+1) respectively, the voltages U_(ci) and U_(c,i+1) between the output points of the filter at the instants t_(i) and t_(i+1) respectively, the average line voltage Ū_(l) between the instants t_(i) and t_(i+1) and the average intensity Ī_(u); and controlling the electric converter to produce a current I_(u) flowing through the output point, the average intensity Ī_(u) of which between the instants t_(i) and t_(i+1) is equal to the setting Ī_(uc), the time interval T between the instants t_(i) and t_(i+1) being strictly less than 5τ.
 10. The method according to claim 9, wherein the line current setting I_(lc) is chosen always to be less than or equal to a limit I_(lmax), the limit I_(lmax) corresponding to the intensity of the line current I_(l) at which a circuit-breaker of a power substation of the catenary or of the electric vehicle is tripped.
 11. The method according to claim 9, wherein the line current setting I_(lc) is chosen always to be greater than or equal to a limit I_(lmin) the limit I_(lmin) corresponding to the intensity of the line current below which the inductance L of the filter is desaturated.
 12. The method according to claim 9, wherein the method includes: constructing an estimate U_(cp) of the voltage U_(c) which will be reached between the output points of the filter at the instant t_(i+1) if the intensity of the line current I_(l) is equal to the line current setting I_(lc) at the instant t_(i+1); comparing the voltage estimate U_(cp) with at least one predetermined voltage limit U_(cm); only if the predetermined voltage limit U_(cm) is crossed, modifying the line current setting I_(lc) so as to obtain a temporary line current setting I_(lcm) which corresponds to a voltage estimate U_(cp) that does not cross the predetermined voltage limit U_(cm), and using the temporary line current setting I_(lcm) in place of the line current setting I_(lc) during the control of the converter only over the interval T in progress; and if the predetermined voltage limit U_(cm) is not crossed, using the line current setting I_(lc) during the calculation of the current setting Ī_(uc) for the interval T in progress.
 13. The method according to claim 9, wherein the current setting Ī_(uc) is a solution of the following system of equations: $\begin{matrix} {{I_{lc} - {\mu_{2} \cdot C \cdot U_{cp}}} = {{{\mathbb{e}}^{\mu_{1} \cdot T} \cdot \left( {I_{li} - {\mu_{2} \cdot C \cdot U_{ci}}} \right)} + {a_{1} \cdot \left( {{\mu_{2} \cdot {\overset{\_}{I}}_{uc}} + {\frac{1}{L} \cdot {\overset{\_}{U}}_{l}}} \right)}}} \\ {{I_{lc} - {\mu_{1} \cdot C \cdot U_{cp}}} = {{{\mathbb{e}}^{\mu_{2} \cdot T} \cdot \left( {I_{li} - {\mu_{1} \cdot C \cdot U_{ci}}} \right)} + {a_{2} \cdot \left( {{\mu_{1} \cdot {\overset{\_}{I}}_{uc}} + {\frac{1}{L} \cdot {\overset{\_}{U}}_{l}}} \right)}}} \end{matrix}$ where: R and L are the values of the resistance and the inductance, respectively, of the RLC filter, and which are connected in series between the first input and output points; C is the capacitance of the capacitor connected between the first and second output points; μ₁ and μ₂ are the eigenvalues of an evolution matrix and are defined by the following relationships: $\begin{matrix} {\mu_{1} = \frac{{{- R} \cdot C} + \sqrt{{R^{2} \cdot C^{2}} - {4 \cdot L \cdot C}}}{2 \cdot L \cdot C}} \\ {{\mu_{2} = \frac{{{- R} \cdot C} - \sqrt{{R^{2} \cdot C^{2}} - {4 \cdot L \cdot C}}}{2 \cdot L \cdot C}};{and}} \end{matrix}$ a₁ and a₂ are values defined by the following relationships: $\begin{matrix} {a_{1} = \frac{{\mathbb{e}}^{\mu_{1} \cdot T} - 1}{\mu_{1}}} \\ {a_{2} = {\frac{{\mathbb{e}}^{\mu_{2} \cdot T} - 1}{\mu_{2}}.}} \end{matrix}$
 14. A method comprising: a first phase for regulating only a voltage U_(c) between first and second output points of a low-pass RLC filter of natural period T_(f), the RLC filter including two input points electrically connected, respectively, to conductors of a DC bus of an electric vehicle powered via a catenary, first and second output points being electrically connected to a controllable electric converter for controlling torque exerted by an electric traction motor of the electric vehicle, stator time constant τ of the electric traction motor being strictly less than the natural period T_(f), the first phase method including measuring or estimating the intensity I_(li) of a line current I_(l) flowing through an inductance of the filter at an instant t_(i), of a voltage U_(ci) between the first and second output points of the filter at the instant t_(i), and of a line voltage U_(l) between the input points of the filter, wherein the first phase method is a deadbeat control method comprising: calculating a setting Ī_(uc) for an average intensity Ī_(u) of a DC current I_(u) flowing through the first output point of the filter between the instant t_(i) and an instant t_(i+1), the setting Ī_(uc) being established from discretized state equations of the filter in such a way that the voltage U_(c) is equal to a predetermined voltage setting U_(cc) at the instant t_(i+1), the discretized state equations between them establishing relationships between the intensities I_(li) and I_(l,i+1) of a line current I_(l) at the instants t_(i) and t_(i+1) respectively, the voltages U_(ci) and U_(c,i+1) between the first and second output points of the filter at the instants t_(i) and t_(i+1) respectively, the average line voltage Ū_(l) between the instants t_(i) and t_(i+1) and the average intensity Ī_(u); controlling the electric converter in order to produce a current I_(u) flowing through the first output point of the filter, the average intensity Ī_(u) of which between the instants t_(i) and t_(i+1) is equal to the setting Ī_(uc), the time interval T between the instants t_(i) and t_(i+1) being strictly less than 5τ; a second phase for regulating only the intensity of the line current I_(l) in accordance with claim 9; and switchover from the first phase to the second phase as soon as the intensity of the line current I_(l) crosses a limit I_(lm) and switchover from the second phase to the first phase as soon as the intensity of the line current I_(l) crosses the same or another limit in the opposite direction.
 15. An electric vehicle including: a DC bus formed by two conductors; at least one traction motor of the electric vehicle having a stator time constant τ; a controllable electric converter intended to cause the torque of the traction motor to vary; a low-pass RLC filter including two input points electrically connected, respectively, to the two conductors of the DC bus, and first and second output points electrically connected to the electric converter; sensors or estimators intended to measure or estimate the intensity I_(li) of a line current I_(l) flowing through the inductance of the filter at the instant t_(i), a voltage U_(ci) between the output points of the filter at the instant t_(i), and a line voltage U_(l) between the input points of the filter; a computer for calculating a current setting Ī_(uc) for the average intensity Ī_(u) of a DC current I_(u) flowing through the first output point between the instant t_(i) and an instant t_(i+1), this current setting Ī_(uc) being established from discretized state equations of the filter in such a way that the voltage U_(c) is equal to a predetermined voltage setting U_(cc) at the instant t_(i+1), the discretized state equations between them establishing relationships between the intensities I_(li); and I_(l,i+1) of the line current I_(l) at the instants t_(i) and t_(i+1) respectively, the voltages U_(ci) and U_(c,i+1) between the output points of the filter at the instants t_(i) and t_(i+1) respectively, the average line voltage Ū_(l) between the instants t_(i) and t_(i+1) and the average intensity Ī_(u); and a control unit for controlling the electric converter in order to produce a current I_(u) flowing through the output point of the filter, the average intensity Ī_(u) of which between the instants t_(i) and t_(i+1) is equal to the setting Ī_(uc), the time interval T between the instants t_(i) and t_(i+1) being strictly less than 5τ.
 16. An electric vehicle including: a DC bus formed by two conductors; at least one traction motor of the electric vehicle having a stator time constant τ; a controllable electric converter intended to cause the torque of the traction motor to vary; a low-pass RLC filter including two input points electrically connected, respectively, to the two conductors of the DC bus, and first and second output points electrically connected to the electric converter; sensors or estimators intended to measure or estimate the intensity I_(li) of a line current I_(l) flowing through the inductance (L) of the filter at the instant t_(i), a voltage U_(ci) between the output points of the filter at the instant t_(i), a voltage U_(l) between the input points of the filter; a computer for calculating a current setting Ī_(uc) for the average intensity Ī_(u) of a DC current I_(u) flowing through the first output point of the filter between the instant t; and an instant t_(i+1), this current setting Ī_(uc) being established from discretized state equations of the filter in such a way that the intensity of the line current I_(l) is equal to a predetermined line current setting I_(lc) at the instant t_(i+1), these discretized state equations between them establishing relationships between the intensities and I_(li) and I_(l,i+1) of the line current I_(l) at the instants t_(i) and t_(i+1) respectively, the voltages U_(ci) and U_(c,i+1) between the output points of the filter at the instants t_(i) and t_(i+1) respectively, the average line voltage Ū_(l) between the instants t_(i) and t_(i+1) and the average intensity Ī_(u); and a control unit for controlling the electric converter in order to produce a current I_(u) flowing through the output point of the filter, the average intensity Ī_(u) of which between the instants t_(i) and t_(i+1) is equal to the current setting Ī_(uc), the time interval T between the instants t_(i) and t_(i+1) being strictly less than 5τ. 